Math 6510 • Differentiable Manifolds

Let \(X\) be a path connected, locally path connected, and semi-locally simply
connected space. (Recall that \(X\) is semi-locally simply connected if every point
in \(X\) has a neighborhood \(U\) such that every closed loop in \(U\) can be contracted
to a point in \(X\).) Outline the construction of the universal cover
\(p:\widetilde{X}\to X\). Specifically,
\begin{enumerate}
\item[(a)] Define \(\widetilde{X}\) as a set and the projection
\(p:\widetilde{X}\to X\),
\item[(b)] Define the topology on \(\widetilde{X}\) (you don't have to check that
this is a topology),
\item[(c)] Give an argument that \(\widetilde{X}\) is simply connected.
\end{enumerate}
Indicate where each of the three assumptions on \(X\) is used.